Summary: We consider the inverse conductivity problem for the equation \[ \text{div} ((1+(k-1)\chi_D)\nabla u)=0 \] determining the unknown object \(D\) contained in a domain \(\Omega\) with one measurement on \(\partial\Omega\). The method in this paper is the layer potential technique. We find a representation formula for the solution to the equation using single layer potentials on \(D\) and \(\Omega\). Using this representation formula, we prove that the location and size of a disk \(D\) contained in a simply connected bounded Lipschitz domain \(\Omega\) can be determined with one measurement corresponding to arbitrary non-zero Neumann data on \(\partial\Omega\). (Previously, it was known that a disk can be determined with one measurement if \(\Omega\) is assumed to be the half space.) We also prove a weaker version of the uniqueness for balls in \(\mathbb{R}^n\) \((n\geq 3)\) with one measurement corresponding to a certain Neumann data.